User:IssaRice/Distribution of X over Y

A distribution of $X$ over $Y$ is any function $f:Y\to X$.

Examples:

• A probability distribution over a finite sample space is a distribution of probabilities $[0,1]$ over the sample space $\mathrm{\Omega }$. In other words, a probability distribution is a function $p:\mathrm{\Omega }\to [0,1]$. In the case of probability distributions, we also require some extra conditions (namely that ${\sum }_{\omega \in \mathrm{\Omega }}p(\omega )=1$).
• A wealth distribution over people is a distribution of wealth ($\mathbf{R}$) over a set of people ($P$), that is, a function $w:P\to \mathbf{R}$, where $P$ is a set of people. Thus, if $x\in P$ then $w(x)$ is $x$'s wealth.
• A gender distribution over people assigns a gender to each person. $g:P\to G$
• A gender distribution can also be thought of as a distribution of sets of people over genders. $g^{\prime }:G\to 2^P$ where $g^{\prime }(x)=g^{-1}(\{x\})=\{p\in P:g(p)=x\}$

The definition above runs into trouble when the set $Y$ is uncountable. In this case, we might not be able to find any function that satisfies the extra conditions we want to place on the distribution (e.g. in the case of probability distributions, we want to assign 0 to every individual outcome, but then the sum is also 0 rather than 1).

It seems like one way to get around this is to change the type of a distribution. Namely, rather than a function $f:Y\to X$ we have some collection $\mathcal{F}\subseteq 2^Y$ of subsets of $Y$, and we define a function $f:\mathcal{F}\to X$. What constraints should this new function satisfy? In the case of probability, we have a condition that translates between subsets and real numbers.